The linear FEAST algorithm is a method for solving linear eigenvalue problems. It uses complex contour integration to calculate the eigenvectors whose eigenvalues that are located inside some user-defined region in the complex plane. This makes it possible to parallelize the process of solving eigenvalue problems by simply dividing the complex plane into a collection of disjoint regions and calculating the eigenpairs in each region independently of the eigenpairs in the other regions. In this paper we present a generalization of the linear FEAST algorithm that can be used to solve nonlinear eigenvalue problems. Like its linear progenitor, the nonlinear FEAST algorithm can be used to solve nonlinear eigenvalue problems for the eigenpairs whose eigenvalues lie in a user-defined region in the complex plane, thereby allowing for the calculation of large numbers of eigenpairs in parallel. We describe the nonlinear FEAST algorithm, and use several physically-motivated examples to demonstrate its properties.

Abstract. In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given m × n matrix A by a matrix B of rank at most k which is much smaller than m and n. The best rank k approximation can be determined via the singular value decomposition which, however, has prohibitively high computational complexity and storage requirements for very large m and n.We present an optimal least squares algorithm for computing a rank k approximation to an m×n matrix A by reading only a limited number of rows and columns of A. The algorithm has complexity O(k 2 max(m, n)) and allows to iteratively improve given rank k approximations by reading additional rows and columns of A. We also show how this approach can be extended to tensors and present numerical results.

This paper presents adaptive algorithms for eigenvalue problems associated with non-selfadjoint partial differential operators. The basis for the developed algorithms is a homotopy method which departs from a well-understood selfadjoint problem. Apart from the adaptive grid refinement, the progress of the homotopy as well as the solution of the iterative method are adapted to balance the contributions of the different error sources. The first algorithm balances the homotopy, discretization and approximation errors with respect to a fixed stepsize τ in the homotopy. The second algorithm combines the adaptive stepsize control for the homotopy with an adaptation in space that ensures an error below a fixed tolerance ε. The outcome of the analysis leads to the third algorithm which allows the complete adaptivity in

We consider a new adaptive finite element (AFEM) algorithm for self-adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finitedimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than the previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy.KEY WORDS: eigenvalue problem; finite element method (FEM); adaptive finite element method (AFEM); elliptic eigenvalue problem; Krylov subspace method; error estimate; discretization error; approximation error; roundoff error * Correspondence to: Volker Mehrmann,

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